A dark spacetime grid stretches silently across a cosmic void. As a glowing sphere glides through it, the grid bends and ripples around its path, visualizing the curvature of spacetime from Einstein’s General Relativity. Subtle 3D rotation and faint starlight enhance the scene’s depth, revealing gravity as the graceful warping of the universe’s fabric.
5 Minute Guide to ERP3
Information technology has transformed the way we live and the way we do business. ERP, or Enterprise Resource Planning, is one of the most widely implemented business software systems in a wide variety of industries and organizations. In this short article, we’ll try…
Even & Odd Functions
A function is said to be an even function if the sign of the image does not change when the sign of the preimage changes. Conversely, a function is called an odd function when the sign of the image changes when the sign of the preimage changes. For Even functions,…
Characteristics of a chemical reaction.
A chemical reaction generally has one or more of the below-mentioned characteristics. Change in state Change in colour Evolution of gas Change in temperature Appearance of light Formation of Precipitate 1> Change in stateCertain chemical reactions are featured with a…
What is a Chemical Reaction?
A process in which one or more substances get transformed to produce new substance or substances is called a Chemical Reaction. A chemical reaction involves changes in the position of electrons of atoms by restructuring chemical bonds, with no changes to the nuclei of…
Probability Cause and Effect Problem
Question What does it mean for one event 𝐶 to cause another event 𝐸 – for example, smoking (𝐶) to cause cancer (𝐸)? There is a long history in philosophy, statistics, and the sciences of trying to clearly analyze the concept of a cause. One tradition says that…
Probability Problem: Suppose you roll a fair die two times. Let 𝐴 be the event “THE SUM OF THE THROWS EQUALS 5” and 𝐵 be the event “AT LEAST ONE OF THE THROWS IS A 4”. Solve for the probability that the sum of the throws equals 5, given that at least one of the throws is a 4. That is, solve 𝑃(𝐴|𝐵).
Solution We have A = (1,4), (2,3), (3,2), (4,1) B = (1,4), (2,4), (3,4), (4,4), (5,4), (6,4), (4,1), (4,2), (4,3), (4,5), (4,6) \( P(A|B) = \dfrac {P(A∩B)}{P(B)} \) \( A∩B = (1,4), (4,1) \) The sample space comprises of 6×6 = 36 eventsHence,\( P(A∩B) = \dfrac{2}{36} =…
THEOREM# \( \lim_{\theta\to0} \dfrac{sinθ}{θ} \) = 1
We have \( \lim_{\theta\to0} { \sin\theta \over \theta } \) = 1 Consider the below diagram. We have r = radius of the circle.A = centre of the circle.The sector ⌔ formed by the arc BD subtends an angle θ at the centre. Case 1 : θ > 0 i.e. θ is +ve Let 0 ≤ θ ≤ \(…
Theorem# \( \lim_{x \to a} { x^n – a^n \over x – a } = na^{n-1} \)
To prove : lim\( _{x \to a} { x^n – a^n \over x – a } = na^{n-1} \) where n is a rational number Proof: Let \( x = a + h \) Then as \(x \to a \), we have \(h \to 0 \) Now, \( \lim_{x \to a} { x^n – a^n \over x – a }…
Theorem# Limit of tanθ as θ → 0
Proof : We have, lim\(_{θ\to 0} { \dfrac {\mathrm tan \mathrm θ}{ \mathrm θ} } \) = lim\(_{θ\to 0} { \dfrac {\mathrm \sin \mathrm θ} {\mathrm θ \mathrm \cos\mathrm θ} } \) \( \{∵ \tan\theta = \dfrac…
Theorem# Limit of cosθ as θ → 0
As θ → 0, we have cosθ → 1 Proof : When θ = 0, We have, lim\(_{θ\to 0} \cos \)θ = cos0 = 1 { ∵ cos0 = 1 } Hence, lim\(_{θ\to 0} \cos \)θ = 1